We establish uniform bounds for the solutions $e^{ it\Delta}u$ of the Schr\"{ o}dinger equation on arithmetic flat tori, generalising earlier results by J. Bourgain. We also study the regularity properties of weak-* limits of sequences of densities of the form $|e^{ it\Delta}u_{ n}|^{ 2}$ corresponding to highly oscillating sequences of initial data $(u_{ n})$. We obtain improved regularity properties of those limits using previous results by N. Anantharaman and F. Maci\`a on the structure of semiclassical measures for solutions to the Schr\"{ o}dinger equation on the torus.
Categories: math.AP math.CA